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Claim. In a blog post Erdos problem #385, the parity problem, and Siegel zeroes (19 August 2024), Terence Tao observes that if, for some $2<u<3$, the largest gap between consecutive semiprimes in $[x,2x]$ whose prime factors lie in $[x^{1/u},x^{1-1/u}]$ is $o(x^{1/u})$, then both questions of Problem 385 have the answer yes. Indeed, such a semiprime $m<n$ within $o(n^{1/u})$ of $n$ has least prime factor at least $(n/2)^{1/u}$, so $m+p(m)-n\to\infty$, which gives $F(n)>n$ for all large $n$ and $F(n)-n\to\infty$.

Hypothesis. The gap bound is unproved. Tao calls it very plausible, as a consequence of a semiprime analog of Cramér's conjecture, but out of reach: even on the Riemann hypothesis, gaps between primes in $[x,2x]$ are bounded only by $O(\sqrt x\log x)$, and the bound for semiprimes is not much better, falling short of $x^{1/u}$ for every $2<u<3$. Tao argues that the parity problem blocks sieve methods here and expects no resolution without a breakthrough strong enough to exclude Siegel zeros.

Acceptance. None. The post is not refereed, and the site's commentary, on a problem it labels OPEN, only mentions it.

Depends on. No page of this wiki.